\chapter{—‰‘¤ş‰¿‰»‰‚ı õ‰Æ‰‘ş‰Û î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â \label{history}}
õ‰Æ‰ÿ‰Ü‰‚ı •‰ƒ‰À� î‰Â¢ö î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â ¬‰�¤–û‰‘ı õ‰µ‰Ô‰‘ø—‰ü ¢�¤¢. 
¨‰‘¢ù—‰Âş‰ß Ÿ‰‘ó‰´ �ş‰ß õ‰Æ‰ÿ‰Ü‰‚, •‰ƒ‰À� î‰Â¢ö î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â “‰ƒ‰ß ¢ø ÷‰Ö‰Î‰‚ ¢¤ ¢�¡‰Û
ş‰× �‰€‰À®‰Ü‰ã‰ü ¨‰‘¢ù �¨‰´. ¢¤ �ş‰ß Ÿ‰‘ó‰´, õ‰ü—‰��÷‰ƒ‰İ “‰‘ �¨‰µ‰Ô‰‘¢ù �¥ õ‰·‰Ü‰¶“‰€‰Àı î‰Â¢ö
�‰€‰À®‰Ü‰ã‰ü, •‰ƒ‰À� î‰Â¢ö ¢÷‰±‰‘ó‰‚ı õ‰·‰Ü‰¶û‰‘ş‰ü  î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â “‰‘ş‰À �¥ �öû‰‘ 䉱‰�¤ ‰À,
ø ¨‰‘¡‰µ‰ß î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â “‰‘ �¨‰µ‰Ô‰‘¢ù �¥ �ş‰ß ¢÷‰±‰‘ó‰‚, �ó‰Ú‰�¤ş‰µ‰Ş‰ü �¥ \InE{}$O(n)$\EnE{}
“‰Â�ı �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ “‰‚ ¢¨‰´ �ø¤¢. “‰Â�ı “‰‚ ¢¨‰´ �ø¤¢ö �ω…䉑– “‰ƒ‰Ç—‰Â ¢¤“‰‘¤ùı 
�ş‰ß õ‰Æ‰ÿ‰Ü‰‚ ø õ‰Æ‰‘ş‰Û õ‰Â“‰�¯ “‰‚ �ö, õ‰ü—‰��÷‰ƒ‰À “‰‚ õ‰Â›‰â \cite{survey}
õ‰Â�›‰ã‰‚ ‰ƒ‰À.

Ÿ‰‘ó‰´ •‰ƒ‰»‰ƒ‰Àù—‰Â õ‰Æ‰ÿ‰Ü‰‚ı î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â ¢ø“‰ã‰Àı, 
Ÿ‰‘󉵉ü �¨‰´  õ‰��÷‰ã‰ü “‰‚ ª‰Ø‰Û �‰€‰À®‰Ü‰ã‰ü ¢�ª‰µ‰‚ “‰‘ª‰ƒ‰İ  ÷‰µ‰��÷‰ƒ‰İ �¥ ¢�¡‰Û �öû‰‘ 䉱‰�¤
‰ƒ‰İ. ¢¤ �ş‰ß Ÿ‰‘ó‰´ “‰‘ ¨‰‘¡‰µ‰ß ş‰× 
ğ‰Â�é ¢ş‰À\footnote{\InE{}Visibility Graph\EnE{}} ø �¨‰µ‰Ô‰‘¢ù �¥ ş‰Ø‰ü �¥ �ó‰Ú‰�¤ş‰µ‰İû‰‘ı •‰ƒ‰À� 
î‰Â¢ö î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â ¢¤ ğ‰Â�éû‰‘ õ‰‘÷‰€‰À �ó‰Ú‰�¤ş‰µ‰İ \InE{}Dijkstra\EnE{} õ‰ü—‰��ö õ‰Æ‰ÿ‰Ü‰‚ ¤�
Ÿ‰Û î‰Â¢. “‰‘ �¨‰µ‰Ô‰‘¢ù �¥ �ş‰ß —‰Ø‰€‰ƒ‰×, õ‰ü—‰��ö �ó‰Ú‰�¤ş‰µ‰Ş‰ü �¥ õ‰Â—‰±‰‚ı \InE{}$O(E_{VG}+n\log n)$\EnE{}
“‰Â�ı �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ •‰ƒ‰À� î‰Â¢  õ‰€‰Ñ‰�¤ �¥ \InE{}$E_{VG}$\EnE{} —‰ã‰À�¢ ş‰‘ñû‰‘ı ğ‰Â�é ¢ş‰À �¨‰´.
“‰€‰‘“‰Â�ş‰ß ¢¤ “‰À—‰Âş‰ß Ÿ‰‘ó‰´ ¥õ‰‘ö �›‰Â�ı �ş‰ß �ó‰Ú‰�¤ş‰µ‰İ “‰ÂŸ‰Æ‰°  \InE{}$n$\EnE{} �¥ \InE{}$O(n^2)$\EnE{} �¨‰´.
¤ø© ¢ş‰Ú‰Âı  “‰Â�ı �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ ø “‰Æ‰ƒ‰‘¤ı �¥ ¢ş‰Ú‰Â õ‰Æ‰‘�‰Û î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â 
õ‰�¤¢ �¨‰µ‰Ô‰‘¢ù ø�ì‰â ª‰Àù �¨‰´, ¤ø© \InE{}Dijkstra\EnE{}ı •‰ƒ‰�¨‰µ‰‚\footnote{\InE{}Continuous Dijkstra\EnE{}}
�¨‰´. ¢¤ �ş‰ß ¤ø© “‰‚ ›‰‘ı �ş‰ß  �“‰µ‰À� ğ‰Â�é ¢ş‰À ¤� “‰Æ‰‘¥ş‰İ ø ¨‰³‰Å ¤øı �ö 
“‰‚ ¢÷‰±‰‘ñ î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â ›‰Æ‰µ‰¹‰� ‰ƒ‰İ, õ‰Æ‰µ‰Ö‰ƒ‰Ş‰\nasb ‘ ÷‰Ö‰È‰‚ı î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Âû‰‘
¤� õ‰ü¨‰‘¥ş‰İ. “‰ú‰µ‰Âş‰ß 
�ó‰Ú‰�¤ş‰µ‰Ş‰ü  —‰‘ ‰�ö “‰Â�ı Ÿ‰Û õ‰Æ‰ÿ‰Ü‰‚ı î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â “‰‘  õ‰��÷‰â
�‰€‰À®‰Ü‰ã‰ü “‰‘  �¨‰µ‰Ô‰‘¢ù �¥ �ş‰ß ¤ø© �¤��‰‚ ª‰Àù �¨‰´, �¥ ¥õ‰‘ö �›‰Â�ş‰ü �¥
õ‰Â—‰±‰‚ı \InE{}$O(n\log n)$\EnE{} “‰Â¡‰�¤¢�¤ �¨‰´.  �ş‰ß �ó‰Ú‰�¤ş‰µ‰İ —‰�¨‰Í \InE{}Hershberger\EnE{} ø 
\InE{}Suri\EnE{} ¢¤ \cite{sref83, sref84} �¤��‰‚ ª‰Àù  �¨‰´.

Ÿ‰‘ó‰´û‰‘ı ¢ş‰Ú‰Âı ÷‰ƒ‰Ã “‰Â�ı õ‰Æ‰ÿ‰Ü‰‚ı î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â ¢¤ ê‰Ì‰‘ı ¢ø“‰ã‰Àı õ‰Î‰Â�
ª‰Àù �¨‰´. “‰Â�ı õ‰·‰‘ñ õ‰ü—‰��÷‰ƒ‰İ “‰‚ ›‰‘ı õ‰µ‰Âş‰× �ì‰Ü‰ƒ‰À¨‰ü “‰Â�ı ꉑ¬‰Ü‰‚ õ‰µ‰Âş‰×
¢ş‰Ú‰Âı õ‰‘÷‰€‰À õ‰µ‰Âş‰× \InE{}$L_1$\EnE{} ) ¢¤ �ö ꉑ¬‰Ü‰‚ı ¢ø ÷‰Ö‰Î‰‚ı  \InE{}$(x_1, y_1)$\EnE{}
ø \InE{}$(x_2, y_2)$\EnE{} “‰Â�“‰Â “‰‘ \InE{}$|x_1-x_2|+|y_1-y_2|$\EnE{} �¨‰´( ¤�  ¢¤ ÷‰Ñ‰Â
“‰Ú‰ƒ‰Âş‰İ. û‰Ş‰»‰€‰ƒ‰ß õ‰ü—‰��ö õ‰Æ‰ÿ‰Ü‰‚ ¤� “‰Â�ı Ÿ‰‘󉵉ü  õ‰ü¡‰��û‰ƒ‰İ —‰ã‰À�¢  ğ‰Â¢©û‰‘
¢¤ õ‰Æ‰ƒ‰Â õ‰ƒ‰€‰ƒ‰Ş‰İ ª‰�¢ õ‰Î‰Â� ‰ƒ‰İ. Ÿ‰‘ó‰´û‰‘ş‰ü  ¢¤ �ö ÷‰‘Ÿ‰ƒ‰‚û‰‘ı õ‰¿‰µ‰Ü‰Ó ¢�¤�ı 
ø¥öû‰‘ı õ‰¿‰µ‰Ü‰Ó  û‰Æ‰µ‰€‰À ÷‰ƒ‰Ã õ‰�¤¢ “‰Â¤¨‰ü ì‰Â�¤ ğ‰Âꉵ‰‚ �¨‰´. ş‰× ¬‰�¤– ›‰‘ó‰° ¢ş‰Ú‰Â �¥ 
�ş‰ß õ‰Æ‰ÿ‰Ü‰‚, •‰ƒ‰À� î‰Â¢ö õ‰Æ‰ƒ‰Âı “‰ƒ‰ß ¢ø ÷‰Ö‰Î‰‚ �¨‰´  ωü î‰Â¢ö �ö “‰‘ �Ÿ‰µ‰Æ‰‘’ 
¥õ‰‘÷‰ü  “‰Â�ı ª‰Âøá “‰‚ Ÿ‰Â, õ‰µ‰�ì‰Ó ª‰Àö, ø ğ‰Â¢© î‰Â¢ö ¬‰Âé õ‰üª‰�¢ ¥õ‰‘ö ‰µ‰Âı
“‰Ú‰ƒ‰Â¢. ¢¤ �ş‰ß Ÿ‰‘ó‰´ Ÿ‰Û î‰Â¢ö õ‰Æ‰ÿ‰Ü‰‚ “‰‚ Ÿ‰Û õ‰ã‰‘¢„– ¢ş‰Ô‰Â�÷‰Æ‰ƒ‰Û ¡‰‘¬‰ü “‰Âõ‰üğ‰Â¢¢.
䉅øù “‰Â �ş‰ß, õ‰ü—‰��ö õ‰Æ‰ÿ‰Ü‰‚ ¤� ¢¤ Ÿ‰‘󉵉ü  �‰€‰À ª‰Â¯ “‰Â�ı õ‰Æ‰ƒ‰Â ¢�¤ş‰İ õ‰Î‰Â�
î‰Â¢. )“‰Â�ı õ‰·‰‘ñ õ‰Ş‰Ø‰ß �¨‰´ “‰¿‰��û‰ƒ‰İ õ‰Æ‰ƒ‰Âı •‰ƒ‰À� ‰ƒ‰İ  ¢¤ ¢¤›‰‚ı �øñ —‰ã‰À�¢
ğ‰Â¢©û‰‘ ¢¤ �ö õ‰ƒ‰€‰ƒ‰Ş‰İ “‰‘ª‰À, ø ¢¤ ¢¤›‰‚ı “‰ã‰À, ω�ñ �ö õ‰ƒ‰€‰ƒ‰Ş‰İ “‰‘ª‰À.( û‰Ş‰‚ı �ş‰ß
Ÿ‰‘ó‰´û‰‘ ø û‰Ş‰»‰€‰ƒ‰ß Ÿ‰‘ó‰´û‰‘ı ›‰‘ó‰° ¢ş‰Ú‰Âı �¥ �ş‰ß õ‰Æ‰ÿ‰Ü‰‚, ¢�¤�ı ¤“‰Â¢ 䉪‰Ü‰ü û‰Æ‰µ‰€‰À ø 
¢¤ û‰€‰À¨‰‚ı õ‰½‰‘¨‰±‰‘—‰ü õ‰�¤¢ “‰Â¤¨‰ü ì‰Â�¤ ğ‰Âꉵ‰‚�÷‰À. “‰Â�ı õ‰Î‰‘ó‰ã‰‚ı 󉃉Ɖµ‰ü �¥ ¤û‰‘ı
õ‰¿‰µ‰Ü‰Ô‰ü  —‰‘ ‰�ö ¤øı Ÿ‰‘ó‰´û‰‘ı õ‰¿‰µ‰Ü‰Ó �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ �÷‰¹‰‘ô ª‰Àù �¨‰´, õ‰ü—‰��÷‰ƒ‰À “‰‚
õ‰Â›‰â \cite{survey} õ‰Â�›‰ã‰‚ ‰ƒ‰À.

õ‰Æ‰ÿ‰Ü‰‚ı î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â ¢¤ Ÿ‰‘ó‰´ ¨‰‚“‰ã‰Àı ş‰‘ “‰ƒ‰È‰µ‰Â �¥ ¨‰‚ “‰ã‰À, õ‰Æ‰ÿ‰Ü‰‚ı “‰Æ‰ƒ‰‘¤ õ‰È‰Ø‰Ü‰ü
�¨‰´. ¢¤  �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ õ‰ü¡‰��û‰ƒ‰İ “‰ƒ‰ß ¢ø ÷‰Ö‰Î‰‚ ¢¤ ê‰Ì‰‘, î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Âı ¤� •‰ƒ‰À�
‰ƒ‰İ  �¥ ¢�¡‰Û “‰ã‰Ì‰ü �¥ õ‰��÷‰â ¨‰‚ “‰ã‰Àı 䉱‰�¤ ÷‰Ø‰€‰€‰À. •‰ƒ‰»‰ƒ‰Àğ‰ü �ş‰ß õ‰Æ‰ÿ‰Ü‰‚, �¥ 
¢ø 䉑õ‰Û ÷‰‘ª‰ü õ‰üª‰�¢. ÷‰¿‰Æ‰µ‰ƒ‰ß 䉑õ‰Û, •‰ƒ‰»‰ƒ‰Àğ‰ü õ‰Æ‰ÿ‰Ü‰‚ �¥ ÷‰Ñ‰Â ›‰±‰Âı �¨‰´.
¨‰‘¡‰µ‰‘¤ î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â ¢¤ �ş‰ß õ‰Æ‰ÿ‰Ü‰‚, ó‰Ãøõ‰ü ÷‰À�¤¢  “‰Â ¤øı û‰ƒ‰º ğ‰Â�é
ğ‰Æ‰Æ‰µ‰‚�ı ø�ì‰â ª‰�¢ )õ‰·‰Û Ÿ‰‘ó‰´ ¢ø“‰ã‰Àı  �ş‰ß õ‰Æ‰ƒ‰Â û‰Ş‰��¤ù ¤øı ğ‰Â�é ¢ş‰À
ø�ì‰â õ‰üª‰À(. ¢¤ Ÿ‰‘ó‰´ ¨‰‚“‰ã‰Àı, î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â “‰‚ ¬‰�¤– ş‰× ¡‰Í ª‰Ø‰Æ‰µ‰‚ �¨‰´
 ÷‰Ö‰‘¯ ª‰Ø‰Æ‰µ‰Ú‰ü �ö ¢¤ Ÿ‰‘ó‰´ î‰Ü‰ü ÷‰Ö‰‘ωü ¢¤ ¢�¡‰Û ş‰‘ñû‰‘ı �‰€‰Àø›‰ú‰üû‰‘ı õ‰‘÷‰â
û‰Æ‰µ‰€‰À ø �¥ �ş‰ß ¡‰‘¬‰ƒ‰´ “‰Â¡‰�¤¢�¤ û‰Æ‰µ‰€‰À  õ‰Æ‰ƒ‰Â “‰‘ş‰À “‰‘ ş‰× ¥�øş‰‚ “‰‚ û‰Â
ş‰‘ñ “‰Â¨‰À ø “‰‘ û‰Ş‰‘ö ¥�øş‰‚ �ö ş‰‘ñ ¤� —‰Âí ‰À; “‰€‰‘“‰Â�ş‰ß ¢¤ ¬‰�¤—‰ü  
õ‰Æ‰ƒ‰Â ¤� ¢¤ �ş‰ß ÷‰Ö‰‘¯ {\em “‰‘¥ ‰ƒ‰İ}, õ‰Æ‰ƒ‰Â —‰±‰Àş‰Û “‰‚ ş‰× ¡‰Í õ‰Æ‰µ‰Ö‰ƒ‰İ õ‰üª‰�¢.
“‰‘ �¨‰µ‰Ô‰‘¢ù �¥ �ş‰ß ¡‰‘¬‰ƒ‰´, ¢¤ ¬‰�¤—‰ü  ş‰× ¢÷‰±‰‘ó‰‚
�¥  ş‰‘ñû‰‘ş‰ü  õ‰Æ‰ƒ‰Â “‰‘ş‰À �¥ �öû‰‘ 䉱‰�¤ ‰À ¤� ¢�ª‰µ‰‚ “‰‘ª‰ƒ‰İ, õ‰ü—‰��ö
õ‰¿‰µ‰Ê‰‘– ÷‰Ö‰‘¯ ª‰Ø‰Æ‰µ‰Ú‰ü ¤� õ‰½‰‘¨‰±‰‚ î‰Â¢ ø “‰‘  �¨‰µ‰Ô‰‘¢ù �¥ �ö ω�ñ î‰�—‰‘ù—‰Âş‰ß
õ‰Æ‰ƒ‰Â ¤� ÷‰ƒ‰Ã “‰‚ ¢¨‰´ �ø¤¢. øó‰ü õ‰È‰Ø‰Û �¥ �ş‰ß›‰‘ ÷‰‘ª‰ü õ‰üª‰�¢  ™‰‘“‰´ ª‰Àù �¨‰´
 õ‰ã‰‘¢„—‰ü  “‰‘ �¨‰µ‰Ô‰‘¢ù �¥ �ş‰ß ¤ø© “‰‚ ¢¨‰´ õ‰ü�ş‰€‰À ø õ‰¿‰µ‰Ê‰‘– ¤�‰�§ õ‰Æ‰ƒ‰Â
¤ş‰È‰‚û‰‘ı �ö õ‰ã‰‘¢„– û‰Æ‰µ‰€‰À, õ‰Ş‰Ø‰ß �¨‰´ õ‰ã‰‘¢„– —‰¹‰Ãş‰‚÷‰‘•‰Áş‰Â �¥ ¢¤›‰‚ı ÷‰Ş‰‘ş‰ü
“‰‘ª‰€‰À. “‰€‰‘“‰Â�ş‰ß õ‰½‰‘¨‰±‰‚ı õ‰¿‰µ‰Ê‰‘– ÷‰Ö‰‘¯ “‰‚ õ‰€‰Ñ‰�¤ õ‰Ö‰‘ş‰Æ‰‚ı ω�ñ õ‰Æ‰ƒ‰Âû‰‘
÷‰ƒ‰‘¥õ‰€‰À �ş‰ß �¨‰´  “‰‚ ş‰× —‰ã‰À�¢ ÷‰Ş‰‘ş‰ü õ‰½‰‘¨‰±‰‚ �÷‰¹‰‘ô ¢û‰ƒ‰İ.

õ‰È‰Ø‰Û ¢øô õ‰Æ‰ÿ‰Ü‰‚ı î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â ¨‰‚“‰ã‰Àı, •‰ƒ‰»‰ƒ‰Àğ‰ü õ‰Æ‰ÿ‰Ü‰‚ �¥ ÷‰Ñ‰Â —‰Â‰±‰ƒ‰‘—‰ü
�¨‰´. —‰ã‰À�¢ Ÿ‰‘ó‰´û‰‘ş‰ü  ¢÷‰±‰‘ó‰‚ı ş‰‘ñû‰‘ş‰ü  î‰�—‰‘ù—‰Âş‰ß  õ‰Æ‰ƒ‰Â �öû‰‘ ¤� ì‰Î‰â
õ‰ü‰À ¢�¤¢, �¥ õ‰Â—‰±‰‚ı ÷‰Ş‰‘ş‰ü �¨‰´. “‰‘ �¨‰µ‰Ô‰‘¢ù �¥  �ş‰ß õ‰�®‰�á, ™‰‘“‰´ ª‰Àù �¨‰´
 õ‰Æ‰ÿ‰Ü‰‚ı î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â ¢¤ Ÿ‰‘ó‰´ ¨‰‚ “‰ã‰Àı, ş‰× õ‰Æ‰ÿ‰Ü‰‚ı \InE{}NP--Hard\EnE{} �¨‰´. )Ÿ‰µ‰‘ 
¢¤ Ÿ‰‘󉵉ü  õ‰��÷‰â, ş‰× õ‰¹‰Ş‰�䉂 �¥ õ‰·‰Ü‰¶û‰‘ı õ‰��¥ı û‰İ “‰‘ª‰€‰À ÷‰ƒ‰Ã ™‰‘“‰´ ª‰Àù �¨‰´
 õ‰Æ‰ÿ‰Ü‰‚ \InE{}NP--Hard\EnE{} �¨‰´.( “‰€‰‘“‰Â�ş‰ß —‰€‰ú‰‘ �ó‰Ú‰�¤ş‰µ‰İû‰‘ı ¢ì‰ƒ‰Ö‰ü 
“‰Â�ı �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ ª‰€‰‘¡‰µ‰‚ ª‰Àù û‰Æ‰µ‰€‰À, �¥ õ‰Â—‰±‰‚ı ÷‰Ş‰‘ş‰ü û‰Æ‰µ‰€‰À.
“‰€‰‘“‰Â�ş‰ß ω±‰ƒ‰ã‰ü �¨‰´  “‰‚ ¢÷‰±‰‘ñ �ó‰Ú‰�¤ş‰µ‰İû‰‘ı —‰Ö‰Âş‰±‰ü “‰Â�ı Ÿ‰‘ó‰´ î‰Ü‰ü
�ş‰ß õ‰Æ‰ÿ‰Ü‰‚, ø ş‰‘ �ó‰Ú‰�¤ş‰µ‰İû‰‘ı  ¢ì‰ƒ‰Õ “‰Â�ı Ÿ‰‘ó‰´û‰‘ı ¡‰‘« �ö  �ó‰Ú‰�¤ş‰µ‰İ 
¨‰Âş‰â ¢�¤÷‰À “‰‘ª‰ƒ‰İ.

�øó‰ƒ‰ß ÷‰µ‰ƒ‰¹‰‚ı î‰Ü‰ü ¢¤ �ş‰ß ¥õ‰ƒ‰€‰‚, �ó‰Ú‰�¤ş‰µ‰İ \InE{}Papadimitriou\EnE{} �¨‰´  ¢¤ ¥õ‰‘÷‰ü
�¥ \InE{}$O(n^4(L+\log({n\over\varepsilon}))^2/\varepsilon^2)$\EnE{}, õ‰Æ‰ƒ‰Âı “‰‘ ω�ñ
Ÿ‰À�‰Â \InE{}$(1+\varepsilon)$\EnE{} “‰Â�“‰Â ω�ñ õ‰Æ‰ƒ‰Â “‰ú‰ƒ‰€‰‚ •‰ƒ‰À� õ‰ü‰À. ¢¤ �ş‰ß›‰‘ \InE{}$L$\EnE{} —‰ã‰À�¢
“‰ƒ‰´û‰‘ı ¢ì‰´ ¢¤ õ‰Àñ õ‰½‰‘¨‰±‰‘—‰ü �¨‰´. •‰Å �¥ �ş‰ß �ó‰Ú‰�¤ş‰µ‰İ,  \InE{}Clarkson\EnE{} �ó‰Ú‰�¤ş‰µ‰Ş‰ü 
“‰Â�ı õ‰½‰‘¨‰±‰‚ı õ‰Æ‰ƒ‰Âı “‰‘ ω�ñ Ÿ‰À�‰Â \InE{}$(1+\varepsilon)$\EnE{} “‰Â�“‰Â ω�ñ õ‰Æ‰ƒ‰Â “‰ú‰ƒ‰€‰‚
¢¤ ¥õ‰‘ö \InE{}$O(n^2\log n/{\varepsilon^4})$\EnE{} �¤��‰‚ ¢�¢. 

ş‰Ø‰ü �¥ Ÿ‰‘ó‰´û‰‘ı ¡‰‘« �ş‰ß õ‰Æ‰ÿ‰Ü‰‚  ¢�¤�ı ¤“‰Â¢ ê‰Â�ø�ö �¨‰´, õ‰Æ‰ÿ‰Ü‰‚ı •‰ƒ‰À� î‰Â¢ö 
î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â ¤øı ¨‰Î‰¼ ş‰× �‰€‰Àø›‰ú‰ü �¨‰´; ş‰ã‰€‰ü õ‰ü¡‰��û‰ƒ‰İ “‰ƒ‰ß ¢ø ÷‰Ö‰Î‰‚ı ¢�¢ù ª‰Àù
¤øı ¨‰Î‰¼ ş‰× �‰€‰Àø›‰ú‰ü, õ‰Æ‰ƒ‰Âı •‰ƒ‰À� ‰ƒ‰İ  �ş‰ß õ‰Æ‰ƒ‰Â “‰‚ ω�¤ õ‰Û ¤øı ¨‰Î‰¼ �‰€‰Àø›‰ú‰ü
ø�ì‰â ª‰�¢. �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ “‰‚ ÷‰‘ô õ‰Æ‰ÿ‰Ü‰‚ı ¦�‰�¢¥ş‰× ğ‰Æ‰Æ‰µ‰‚\footnote{\InE{}Discrete Geodesic Problem\EnE{}}
õ‰ã‰Âøé �¨‰´.
õ‰�®‰�á �¬‰Ü‰ü �ş‰ß •‰‘ş‰‘ö÷‰‘õ‰‚, “‰Â¤¨‰ü �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ ø ¤�ùŸ‰Ûû‰‘ı õ‰¿‰µ‰Ü‰Ó �¤��‰‚ 
ª‰Àù “‰Â�ı Ÿ‰Û �ö �¨‰´.

“‰Â�ı õ‰È‰‘û‰Àùı �ş‰ß  �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ ¢¤ Ÿ‰Ö‰ƒ‰Ö‰´ Ÿ‰‘ó‰´ ¡‰‘¬‰ü �¥ õ‰Æ‰ÿ‰Ü‰‚ı 쉱‰Û
�¨‰´, ê‰ü �¨‰´ —‰�›‰‚ ‰ƒ‰À  ¢¤ ¬‰�¤—‰ü  õ‰��÷‰â ¤� ¢ø —‰‘ “‰Ú‰ƒ‰Âş‰İ; ş‰Ø‰ü ¡‰�¢ �‰€‰Àø›‰ú‰ü
ø ¢ş‰Ú‰Âı õ‰Ø‰Ş‰Û �ö )ş‰ã‰€‰ü �‰€‰Àø›‰ú‰üş‰ü  �›‰µ‰Ş‰‘á �ö “‰‘ �‰€‰Àø›‰ú‰ü �ø󉃉‚ î‰Û ê‰Ì‰‘ ¤� õ‰ü•‰�ª‰‘÷‰À(,
¢¤ �ş‰ß ¬‰�¤– Ÿ‰Û õ‰Æ‰ÿ‰Ü‰‚ı 쉱‰Û ¢¤ Ÿ‰Ö‰ƒ‰Ö‰´ ş‰× Ÿ‰Û “‰Â�ı õ‰Æ‰ÿ‰Ü‰‚ı •‰ƒ‰À� î‰Â¢ö î‰�—‰‘ù—‰Âş‰ß
õ‰Æ‰ƒ‰Â ¤øı ¨‰Î‰¼ �‰€‰Àø›‰ú‰ü �¨‰´.

�¥ ωÂé ¢ş‰Ú‰Â �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ ş‰× Ÿ‰‘ó‰´ ¡‰‘« �¥  õ‰Æ‰ÿ‰Ü‰‚ı •‰ƒ‰À� î‰Â¢ö î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â
¢¤ ş‰× ê‰Ì‰‘ı ¢ø“‰ã‰Àı “‰‘ ø›‰�¢ õ‰��÷‰â �¨‰´. “‰Â�ı õ‰È‰‘û‰Àùı �ş‰ß õ‰�®‰�á, —‰�›‰‚ ‰ƒ‰À
 õ‰ü—‰��÷‰ƒ‰İ �¥ ¤øı ş‰× ÷‰Ş‰�÷‰‚ �¥ õ‰Æ‰ÿ‰Ü‰‚ı î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â ¢ø“‰ã‰Àı, ş‰× ÷‰Ş‰�÷‰‚
“‰Â�ı õ‰Æ‰ÿ‰Ü‰‚ı î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â ¤øı ¨‰Î‰¼ “‰‚ �ş‰ß ¬‰�¤– “‰Æ‰‘¥ş‰İ: ¨‰Î‰¼ ÷‰Ş‰�÷‰‚, ş‰×
¬‰Ô‰½‰‚ �¨‰´  ¢¤ ›‰‘û‰‘ş‰ü  õ‰��÷‰â )¢¤ õ‰Æ‰ÿ‰Ü‰‚ı ¢ø“‰ã‰Àı( ¢¤ �öû‰‘ ì‰Â�¤ ğ‰Âꉵ‰‚�÷‰À, 
�¨‰µ‰��÷‰‚û‰‘ş‰ü “‰‘ �¤—‰Ô‰‘á ¥ş‰‘¢  ì‰Â�¤ ¢�¢ù�ş‰İ. Ÿ‰‘„ î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â ¤øı �ş‰ß ¨‰Î‰¼
“‰ƒ‰ß ¢ø ÷‰Ö‰Î‰‚ı ¢�¢ù ª‰Àù, ¢¤ Ÿ‰Ö‰ƒ‰Ö‰´ î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â ¤øı ¬‰Ô‰½‰‚ı ¢ø“‰ã‰Àı “‰ƒ‰ß �ş‰ß ¢ø÷‰Ö‰Î‰‚
�¨‰´  �¥ õ‰��÷‰â ÷‰ƒ‰Ã 䉱‰�¤ ÷‰Ø‰€‰À. “‰€‰‘“‰Â�ş‰ß “‰‘ Ÿ‰Û ª‰Àö õ‰Æ‰ÿ‰Ü‰‚ı õ‰‘, õ‰Æ‰ÿ‰Ü‰‚ı î‰�—‰‘ù—‰Âş‰ß
õ‰Æ‰ƒ‰Â ¢¤ Ÿ‰‘ó‰´ ¢ø“‰ã‰Àı “‰‘ õ‰��÷‰â ÷‰ƒ‰Ã Ÿ‰Û õ‰üª‰�¢.

�è‰Ü‰° �ó‰Ú‰�¤ş‰µ‰İû‰‘ş‰ü  “‰Â�ı Ÿ‰Û �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ �¤��‰‚ ª‰Àù�÷‰À, �ó‰Ú‰�¤ş‰µ‰İû‰‘ı
•‰ƒ‰À� î‰Â¢ö î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â �¥ ş‰× ¤\hamze �§ �‰€‰Àø›‰ú‰ü, “‰‚ û‰Ş‰‚ı ¤�‰�§ ¢ş‰Ú‰Â û‰Æ‰µ‰€‰À.
“‰Â�ı �ş‰ß õ‰€‰Ñ‰�¤, 
�ó‰Ú‰�¤ş‰µ‰İ “‰‘ş‰À ş‰× ¥ş‰Â—‰Ö‰Æ‰ƒ‰İ\footnote{\InE{}Subdivision\EnE{}} �¥ ¨‰Î‰¼ �‰€‰Àø›‰ú‰ü ¤� •‰ƒ‰À� ‰À
 •‰Å �¥ �ş‰ß “‰µ‰��÷‰ƒ‰İ î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â �¥ ¤\hamze �§ õ‰�¤¢ ÷‰Ñ‰Â “‰‚ û‰Â ¤\hamze �§ ¢ş‰Ú‰Â ¤� 
“‰‘ �¨‰µ‰Ô‰‘¢ù �¥ �ş‰ß ¥ş‰Â—‰Ö‰Æ‰ƒ‰İ “‰‚ ¨‰Â䉴 “‰‚ ¢¨‰´ �ø¤ş‰İ. )�ş‰ß õ‰�®‰�á õ‰È‰‘“‰‚ “‰‘ ¤ı
�¨‰´   �ó‰Ú‰�¤ş‰µ‰İû‰‘ı î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â “‰‘ ş‰× õ‰±‰À\hamze �\footnote{\InE{}Single-Source Shortest Path\EnE{}}
õ‰‘÷‰€‰À �ó‰Ú‰�¤ş‰µ‰İ \InE{}Dijkstra\EnE{} ş‰‘ \InE{}Bellman-Ford\EnE{} ¢¤ ğ‰Â�é �÷‰¹‰‘ô õ‰ü¢û‰€‰À.(
¢¤ Ÿ‰Ö‰ƒ‰Ö‰´ ¢ó‰ƒ‰Û �ş‰ß  õ‰Æ‰ÿ‰Ü‰‚ı •‰ƒ‰À� î‰Â¢ö î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â “‰‘ ş‰× õ‰±‰À\hamze � ø ş‰×
õ‰Ö‰Ê‰À õ‰È‰¿‰É õ‰�¤¢  “‰Â¤¨‰ü ì‰Â�¤ ÷‰Ú‰Âꉵ‰‚ �¨‰´, �ş‰ß �¨‰´  û‰ƒ‰º �ó‰Ú‰�¤ş‰µ‰Ş‰ü “‰‘ ¥õ‰‘ö 
�›‰Â�ş‰ü ‰µ‰Â �¥ ¥õ‰‘ö �›‰Â�ı �ó‰Ú‰�¤ş‰µ‰İû‰‘ş‰ü  î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â ¤� �¥ ş‰× õ‰±‰À\hamze �
õ‰È‰¿‰É  ø “‰‚ û‰Ş‰‚ı ¤�‰�§ ¢ş‰Ú‰Â �‰€‰Àø›‰ú‰ü “‰‚ ¢¨‰´ õ‰ü�ø¤÷‰À, “‰Â�ı �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ ª‰€‰‘¡‰µ‰‚ 
÷‰È‰Àù �¨‰´. )õ‰È‰‘“‰‚ �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ ¢¤ õ‰�¤¢ �ó‰Ú‰�¤ş‰µ‰İû‰‘ı •‰ƒ‰À� î‰Â¢ö î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â
¢¤ ğ‰Â�éû‰‘ ÷‰ƒ‰Ã ø›‰�¢ ¢�¤¢.(

�ş‰ß õ‰Æ‰ÿ‰Ü‰‚ “‰Â�ı �øó‰ƒ‰ß “‰‘¤ ¢¤ Ÿ‰‘󉵉ü  �‰€‰Àø›‰ú‰ü õ‰�¤¢ ÷‰Ñ‰Â ş‰× �‰€‰Àø›‰ú‰ü õ‰½‰À’ “‰‘ª‰À,
—‰�¨‰Í \InE{}Sharir\EnE{} ø \InE{}Schorr\EnE{} õ‰�¤¢ “‰Â¤¨‰ü ì‰Â�¤ ğ‰Âꉴ \cite{ref23} ø �ó‰Ú‰�¤ş‰µ‰Ş‰ü �¥ 
õ‰Â—‰±‰‚ı \InE{}$O(n^3\log n)$\EnE{} “‰Â�ı •‰ƒ‰À� î‰Â¢ö ¥ş‰Â—‰Ö‰Æ‰ƒ‰İ õ‰�¤¢ ÷‰Ñ‰Â �¤��‰‚ ª‰À. •‰Å �¥ •‰ƒ‰À�
î‰Â¢ö �ş‰ß ¥ş‰Â—‰Ö‰Æ‰ƒ‰İ, û‰Â î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Âı ¤� õ‰ü—‰��÷‰ƒ‰İ ¢¤ ¥õ‰‘ö \InE{}$O(k+\log n)$\EnE{} 
\InE{}$k$\EnE{} —‰ã‰À�¢ ş‰‘ñû‰‘ı �ş‰ß î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â �¨‰´ “‰‚ ¢¨‰´ �ø¤ş‰İ.

�ó‰Ú‰�¤ş‰µ‰İ “‰ú‰µ‰Âı “‰Â�ı �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ ¢¤ Ÿ‰‘ó‰´ õ‰½‰À’ —‰�¨‰Í \InE{}Mount\EnE{} �¤��‰‚ ª‰À\cite{ref15}.
�ş‰ß �ó‰Ú‰�¤ş‰µ‰İ ¢¤ ¥õ‰‘ö \InE{}$O(n^2\log n)$\EnE{} ¥ş‰Â—‰Ö‰Æ‰ƒ‰İ õ‰�¤¢ ÷‰Ñ‰Â ¤� •‰ƒ‰À� õ‰ü‰À.

¢¤ Ÿ‰‘󉵉ü  �‰€‰Àø›‰ú‰ü õ‰�¤¢ ÷‰Ñ‰Â õ‰½‰À’ ÷‰±‰‘ª‰À, “‰Â�ı �øó‰ƒ‰ß “‰‘¤ ¢¤ \cite{ref17} 
�ó‰Ú‰�¤ş‰µ‰Ş‰ü �¥ \InE{}$O(n^5)$\EnE{} �¤��‰‚ ª‰À. •‰Å �¥ �ö, �ó‰Ú‰�¤ş‰µ‰Ş‰ü —‰�¨‰Í \InE{}Mitchell\EnE{},
\InE{}Mount\EnE{}, ø \InE{}Papadimitriou\EnE{} ¢¤ \cite{mmp} �¤��‰‚ ª‰À  ¥õ‰‘ö �›‰Â�ı �ö �¥
\InE{}$O(n^2\log n)$\EnE{} “‰�¢.

“‰ú‰µ‰Âş‰ß �ó‰Ú‰�¤ş‰µ‰İ ª‰€‰‘¡‰µ‰‚ ª‰Àù “‰Â�ı �ş‰ß õ‰Æ‰ÿ‰Ü‰‚, û‰İ ¢¤ Ÿ‰‘ó‰´ õ‰½‰À’ ø û‰İ ¢¤ Ÿ‰‘ó‰´
艃‰Âõ‰½‰À’, —‰�¨‰Í \InE{}J.~Chen\EnE{} ø \InE{}Y.~Han\EnE{} �¤��‰‚ ª‰Àù �¨‰´ \cite{paper} ø ¥õ‰‘ö �›‰Â�ı
�ö �¥ õ‰Â—‰±‰‚ı \InE{}$O(n^2)$\EnE{} �¨‰´. �ş‰ß �ó‰Ú‰�¤ş‰µ‰İ ¤� ¢¤ “‰¿‰Ç \ref{alg} —‰�®‰ƒ‰¼ ¡‰��û‰ƒ‰İ 
¢�¢. 䉅øù “‰Â �ş‰ß, �ş‰ß �ó‰Ú‰�¤ş‰µ‰İ “‰‚ 䉀‰��ö “‰¿‰È‰ü �¥ �ş‰ß •‰Âø¦ù •‰ƒ‰‘¢ù¨‰‘¥ı ª‰Àù
�¨‰´ ø  —‰�®‰ƒ‰½‰‘—‰ü ¢¤“‰‘¤ùı •‰ƒ‰‘¢ù¨‰‘¥ı �ö ¢¤ ê‰Ê‰Û \ref{implement} �õ‰Àù �¨‰´.

¢¤ õ‰�¤¢ �ş‰ß õ‰Æ‰ÿ‰Ü‰‚, 䉅øù “‰Â �ó‰Ú‰�¤ş‰µ‰İû‰‘ı ¢ì‰ƒ‰Õ, �ó‰Ú‰�¤ş‰µ‰İû‰‘ı —‰Ö‰Âş‰±‰üş‰ü ÷‰ƒ‰Ã
�¤��‰‚ ª‰Àù �¨‰´. û‰Àé �¥ �ş‰ß �ó‰Ú‰�¤ş‰µ‰İû‰‘, •‰ƒ‰À�  î‰Â¢ö ş‰× ›‰��’ —‰Ö‰Âş‰±‰ü ¢¤ ¥õ‰‘÷‰ü
î‰�—‰‘ù—‰Â �¥ ¥õ‰‘ö �›‰Â�ı “‰ú‰µ‰Âş‰ß �ó‰Ú‰�¤ş‰µ‰İû‰‘ı ¢ì‰ƒ‰Õ ª‰€‰‘¡‰µ‰‚ ª‰Àù �¨‰´. ¢¤ �ş‰ß
¥õ‰ƒ‰€‰‚, ş‰Ø‰ü �¥ �ó‰Ú‰�¤ş‰µ‰İ �¤��‰‚ ª‰Àù, �ó‰Ú‰�¤ş‰µ‰Ş‰ü �¨‰´  —‰�¨‰Í \InE{}Hershberger\EnE{} ø
\InE{}Suri\EnE{} ¢¤ \cite{approx} �¨‰´  �¥ ¥õ‰‘ö �›‰Â�ı \InE{}$O(n)$\EnE{} “‰Â¡‰�¤¢�¤ �¨‰´ ø ›‰��“‰ü
“‰‘ ω�ñ Ÿ‰À�‰Â ¢ø “‰Â�“‰Â ›‰��’ “‰ú‰ƒ‰€‰‚ “‰‚ ¢¨‰´ õ‰ü�ø¤¢. �ş‰ß �ó‰Ú‰�¤ş‰µ‰İ ¤� ¢¤ “‰¿‰Ç
\ref{approxalgsec} ¡‰��û‰ƒ‰À ¢ş‰À. 䉅øù “‰Â �ş‰ß, “‰€‰‘ “‰Â \cite{survey},
�ó‰Ú‰�¤ş‰µ‰Ş‰ü �¥ \InE{}$O(n\log{1\over\varepsilon}+{1\over\varepsilon^3})$\EnE{} ÷‰ƒ‰Ã
“‰Â�ı �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ �¤��‰‚ ª‰Àù �¨‰´  ›‰��“‰ü “‰‚ ω�ñ Ÿ‰À�‰Â \InE{}$1+\varepsilon$\EnE{} “‰Â�“‰Â
ω�ñ ›‰��’ “‰ú‰ƒ‰€‰‚ “‰Â�ı �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ “‰‚ ¢¨‰´ õ‰ü�ø¤¢.
¢¤ �ş‰ß ¥õ‰ƒ‰€‰‚ õ‰ü—‰��÷‰ƒ‰À “‰‚ õ‰Â›‰â \cite{approx1} ÷‰ƒ‰Ã õ‰Â�›‰ã‰‚ ‰ƒ‰À.

䉅øù “‰Â �ş‰ß, �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ ¢¤ Ÿ‰‘ó‰´û‰‘ı ¢ş‰Ú‰Âı ÷‰ƒ‰Ã õ‰Î‰Â� ª‰Àù �¨‰´. “‰Â�ı õ‰·‰‘ñ,
ş‰Ø‰ü �¥ Ÿ‰‘ó‰´û‰‘ı ›‰‘󉱉ü  õ‰ü—‰��ö õ‰Æ‰ÿ‰Ü‰‚ ¤� õ‰Î‰Â� î‰Â¢, Ÿ‰‘ó‰´ {\em ø¥ö¢�¤}
�¨‰´. ¢¤ �ş‰ß õ‰Æ‰ÿ‰Ü‰‚, “‰‚ û‰Â ş‰× �¥ ø›‰‚û‰‘ ş‰× ä‰À¢ “‰‚ 䉀‰��ö ø¥ö �ö  ø›‰‚
÷‰Æ‰±‰´ ¢�¢ù ª‰Àù �¨‰´. û‰Ãş‰€‰‚ı ğ‰Áª‰µ‰ß �¥ û‰Â ø›‰‚ “‰Â�“‰Â “‰‘ ø¥ö �ö ø›‰‚ ®‰Â’ ¢¤
õ‰Æ‰‘ꉵ‰ü �¨‰´  ¢¤ �ö ø›‰‚ ωü õ‰ü‰ƒ‰İ. �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ ¢�¤�ı ¤“‰Â¢û‰‘ı
䉪‰Ü‰ü ê‰Â�ø�÷‰ü ¢¤ \InE{}GIS\EnE{} �¨‰´. “‰Â�ı õ‰·‰‘ñ  õ‰Ş‰Ø‰ß �¨‰´ “‰¿‰��û‰ƒ‰İ “‰ƒ‰ß ¢ø
÷‰Ö‰Î‰‚ �¥ ¨‰Î‰¼ ¥õ‰ƒ‰ß î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â ¤� “‰‚ ¢¨‰´ �ø¤ş‰İ, “‰‘ �ş‰ß ª‰Â¯
 ğ‰Áª‰µ‰ß �¥ �’ “‰Â�ı õ‰‘ ¢�¤�ı û‰Ãş‰€‰‚ı “‰ƒ‰È‰µ‰Âı �¨‰´. )“‰‚ ª‰Ø‰Û \ref{water}
—‰�›‰‚ ‰ƒ‰À.(
�ş‰ß õ‰Æ‰ÿ‰Ü‰‚ ¢¤ \cite{weighted1}
õ‰�¤¢ “‰Â¤¨‰ü ì‰Â�¤ ğ‰Âꉵ‰‚ �¨‰´ ø �ó‰Ú‰�¤ş‰µ‰Ş‰ü �¥ \InE{}$O(n^8\log n)$\EnE{} )ø “‰‘ �¨‰µ‰Ô‰‘¢ù �¥
ê‰Ì‰‘ı \InE{}$O(n^4)$\EnE{}( “‰Â�ı �ö
�¤��‰‚ ª‰Àù �¨‰´. û‰Ş‰»‰€‰ƒ‰ß ¢¤ \cite{weighted} �ó‰Ú‰�¤ş‰µ‰İ —‰Ö‰Âş‰±‰üş‰ü  “‰Æ‰ƒ‰‘¤ 
¨‰‘¢ù �¨‰´ “‰Â�ı �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ �¤��‰‚ ª‰Àù �¨‰´. �ş‰ß �ó‰Ú‰�¤ş‰µ‰İ 
“‰Â �¨‰‘§ ì‰Â�¤ ¢�¢ö —‰ã‰À�¢ı ÷‰Ö‰‘¯ \InE{}Steiner\EnE{} ¤øı ş‰‘ñû‰‘ı �‰€‰Àø›‰ú‰ü 
ø ¨‰³‰Å �¨‰µ‰Ô‰‘¢ù �¥ �ó‰Ú‰�¤ş‰µ‰İ \InE{}Dijkstra\EnE{} “‰Â�ı •‰ƒ‰À� î‰Â¢ö î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â ¢¤ 
ğ‰Â�é ¢ş‰À “‰ƒ‰ß �ş‰ß ÷‰Ö‰Î‰‚û‰‘ 䉪‰Û õ‰ü‰À.
�¨‰µ‰Â�—‰Äıû‰‘ı õ‰¿‰µ‰Ü‰Ô‰ü “‰Â�ı ì‰Â�¤ ¢�¢ö �ş‰ß ÷‰Ö‰Î‰‚û‰‘ ¢¤ �ş‰ß õ‰Â›‰â õ‰�¤¢
“‰Â¤¨‰ü ì‰Â�¤ ğ‰Âꉵ‰‚ �¨‰´. 
¥õ‰‘ö �›‰Â�ı ¢ø �ó‰Ú‰�¤ş‰µ‰İ �¤��‰‚ ª‰Àù ¢¤ �ş‰ß õ‰Ö‰‘ó‰‚ “‰‚ ω�¤ õ‰µ‰�¨‰Í
\InE{}$O(n\log n)$\EnE{} ø ¢¤ “‰À—‰Âş‰ß Ÿ‰‘ó‰´ \InE{}$O(n^5)$\EnE{} ø \InE{}$O(n^3\log n)$\EnE{} �¨‰´.
Ÿ‰‘ó‰´ ¢ş‰Ú‰Âı �¥ �ş‰ß õ‰Æ‰ÿ‰Ü‰‚  ¢¤ �ş‰ß õ‰Ö‰‘ó‰‚ õ‰Î‰Â� ª‰Àù �¨‰´, õ‰Æ‰ÿ‰Ü‰‚ı 
•‰ƒ‰À� î‰Â¢ö î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â “‰ƒ‰ß ¢ø ÷‰Ö‰Î‰‚ “‰‘ ¢¤ ÷‰Ñ‰Â ğ‰Âꉵ‰ß ª‰Â�ş‰Í ¡‰‘¬‰ü
õ‰‘÷‰€‰À Ÿ‰À�‰Â ª‰ƒ‰±‰ü  õ‰Æ‰ƒ‰Â õ‰ü—‰��÷‰À ¢�ª‰µ‰‚ “‰‘ª‰À �¨‰´. �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ ÷‰ƒ‰Ã
¤“‰Â¢û‰‘ı 䉪‰Ü‰ü ê‰Â�ø�÷‰ü ¢�¤¢, �¥ 쉱‰ƒ‰Û •‰ƒ‰À� î‰Â¢ö õ‰Æ‰ƒ‰Â ›‰‘¢ùî‰È‰ü “‰ƒ‰ß ¢ø ÷‰Ö‰Î‰‚
¢¤ î‰�û‰Æ‰µ‰‘ö. “‰‚  ª‰Ø‰Û \ref{slope} —‰�›‰‚ ‰ƒ‰À.

\begin{figure}
\vspace*{8cm}

\caption{î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â “‰‘ ¢¤ ÷‰Ñ‰Â ğ‰Âꉵ‰ß û‰Ãş‰€‰‚ı 䉱‰�¤ �¥ �’}
\label{water}
\end{figure}

\begin{figure}
\vspace*{9cm}

\caption{î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â “‰‘ õ‰½‰Àø¢ş‰´ ¢¤ ª‰ƒ‰° õ‰Æ‰ƒ‰Â}
\label{slope}
\end{figure}

Ÿ‰‘ó‰´ ¢ş‰Ú‰Âı �¥ õ‰Æ‰ÿ‰Ü‰‚ı î‰�—‰‘ù—‰Âş‰ß õ‰Æ‰ƒ‰Â, •‰ƒ‰À� î‰Â¢ö õ‰Æ‰ƒ‰Âı “‰ƒ‰ß ¢ø ÷‰Ö‰Î‰‚ �¨‰´
 õ‰¹‰Ş‰�á �¡‰µ‰…é �¤—‰Ô‰‘á ÷‰Ö‰‘¯ ª‰Ø‰Æ‰µ‰Ú‰ü õ‰µ‰��ó‰ü õ‰Æ‰ƒ‰Â ¢¤ �ö ‰µ‰Âş‰ß ª‰�¢.
¢¤ Ÿ‰Ö‰ƒ‰Ö‰´ ¢¤ �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ “‰‘ş‰À õ‰Æ‰ƒ‰Âı ¤� •‰ƒ‰À� ‰ƒ‰İ  ¢¤ Ÿ‰ƒ‰ß ωü �ö �¥ ÷‰Ö‰Î‰‚ı
“‰‘ �¤—‰Ô‰‘á “‰ƒ‰È‰µ‰Â “‰‚ ÷‰Ö‰Î‰‚ı “‰‘ �¤—‰Ô‰‘á ‰µ‰Â, õ‰Ö‰À�¤ ‰µ‰Âı “‰‚  ¨‰Ş‰´ “‰‘„ Ÿ‰Â
‰ƒ‰İ. Ÿ‰‘ó‰´ ¡‰‘« �ş‰ß õ‰Æ‰ÿ‰Ü‰‚, �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ �¨‰´  õ‰ü¡‰��û‰ƒ‰İ “‰ƒ‰ß ¢ø ÷‰Ö‰Î‰‚ õ‰Æ‰ƒ‰Âı
•‰ƒ‰À� ‰ƒ‰İ  ê‰Ö‰Í ¨‰Â•‰‘ş‰ƒ‰€‰ü “‰‘ª‰À. �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ ÷‰ƒ‰Ã ¢�¤�ı ¤“‰Â¢û‰‘ı ê‰Â�ø�÷‰ü
�¥ 쉱‰ƒ‰Û —‰ã‰ƒ‰ƒ‰ß õ‰Æ‰ƒ‰Â ó‰�ó‰‚î‰È‰ü �’ �¨‰´. �ş‰ß õ‰Æ‰ÿ‰Ü‰‚ ¢¤ õ‰Â›‰â \cite{alps}
õ‰�¤¢ “‰Â¤¨‰ü ì‰Â�¤ ğ‰Âꉵ‰‚ �¨‰´ ø �ó‰Ú‰�¤ş‰µ‰Ş‰ü �¥ \InE{}$O(n\log n)$\EnE{} �¤��‰‚ ª‰Àù �¨‰´
 ¨‰‘¡‰µ‰Ş‰‘ö ¢�¢ù�ış‰ü “‰‘ ÷‰‘ô ÷‰Ö‰È‰‚ı ¨‰Î‰¼ �¤—‰Ô‰‘á\footnote{\InE{}Height Level Map\EnE{}}
¤� õ‰ü¨‰‘¥¢  “‰‘ �¨‰µ‰Ô‰‘¢ù �¥ �ö õ‰ü—‰��ö “‰‚ •‰Â¨‰Çû‰‘ş‰ü �¥
쉱‰ƒ‰Û •‰Â¨‰Çû‰‘ı ê‰�ë ¢¤ ¥õ‰‘ö \InE{}$O(\log n)$\EnE{} •‰‘¨‰¾ ¢�¢. û‰Ş‰»‰€‰ƒ‰ß Ÿ‰‘ó‰´ ¢ş‰Ú‰Âı �¥ �ş‰ß 
õ‰Æ‰ÿ‰Ü‰‚ ÷‰ƒ‰Ã ¢¤ �ş‰ß õ‰Â›‰â “‰‚ �ş‰ß ¬‰�¤– õ‰Î‰Â� ª‰Àù �¨‰´: õ‰ü¡‰��û‰ƒ‰İ “‰ƒ‰ß ¢ø ÷‰Ö‰Î‰‚ı
¢�¢ù ª‰Àù �¥ ş‰× ¨‰Î‰¼, ş‰× õ‰Æ‰ƒ‰Â •‰ƒ‰À� ‰ƒ‰İ  �¤—‰Ô‰‘á �ş‰ß õ‰Æ‰ƒ‰Â û‰ƒ‰ºğ‰‘ù �¥ ş‰×
õ‰Ö‰À�¤ õ‰È‰¿‰É “‰ƒ‰È‰µ‰Â ÷‰È‰�¢. ¨‰‘¡‰µ‰Ş‰‘ö ¢�¢ùı �¤��‰‚ ª‰Àù ¢¤ �ş‰ß  õ‰Ö‰‘ó‰‚ õ‰ü—‰��÷‰À
“‰Â�ı •‰‘¨‰¾ ğ‰Ô‰µ‰ß “‰‚ �ş‰ß •‰Â¨‰Ç ÷‰ƒ‰Ã “‰‚ ¤ ¤ø¢.